Interplay of activity and non-reciprocity in tracer dynamics: From non-equilibrium fluctuation-dissipation to giant diffusion

TL;DR

The paper tackles how nonreciprocal tracer–bath interactions modify transport in active and passive media by deriving an overdamped generalized Langevin equation with a non-Markovian memory kernel and a composite colored noise term. It establishes a generalized nonequilibrium fluctuation-dissipation relation and derives the tracer's mean-squared displacement, showing that long-time diffusivity D_eff depends nonmonotonically on the nonreciprocity parameter χ and can diverge at intermediate χ, yielding giant diffusivity. These predictions are validated by numerical simulations of active Brownian particles, reinforcing the robustness of the effect and its relevance to chase-and-run and predator–prey-like dynamics. The results provide a conceptual and quantitative framework for tuning transport in active matter through nonreciprocal interactions with potential implications for biology and soft-matter engineering.

Abstract

Non-reciprocal interactions play a key role in shaping transport in active and passive systems, giving rise to striking nonequilibrium behavior. Here, we study the dynamics of a tracer -- active or passive -- embedded in a bath of active or passive particles, coupled through non-reciprocal interactions. Starting from the microscopic stochastic dynamics of the full system, we derive an overdamped generalized Langevin equation for the tracer, incorporating a non-Markovian memory kernel that captures bath-mediated correlations. This framework enables us to compute the tracer's velocity and displacement response, derive a generalized nonequilibrium fluctuation-dissipation relation that quantifies deviations from equilibrium behavior, and determine the mean-squared displacement (MSD). We find that while the MSD becomes asymptotically diffusive, the effective diffusivity depends non-monotonically on the degree of non-reciprocity and diverges at an intermediate value. This regime of giant diffusivity provides a generic mechanism for enhanced transport in active soft matter and has direct implications for biological systems exhibiting chase-and-run or predator-prey interactions. Our analytical predictions are supported by numerical simulations of active Brownian particles, highlighting experimentally accessible signatures of non-reciprocal interactions in soft matter.

Figures (2)

• Figure 1: Plots of the nonequilibrium departures $I_v(\omega)$ in (a) and $I_x(\omega)$ in (b) as functions of the nonreciprocal parameter $\chi=\Lambda/\lambda$ for different values of $\omega$. The results are obtained using Eqs. \ref{['eq_velo_depart']} and \ref{['eq_pos_depart']}, respectively. The remaining parameters for both plots are set to $\mathcal{D}_s=\mathcal{D}_b=0.1$ with $\tau_s=\tau_b=0.1$ and $D_{\text{eq}}=0.1$.
• Figure 2: (a) Mean-squared displacement (MSD) scaled by time, $\Delta_s(t)/t$, as a function of $t$ for a tracer attached to a bath particle, for different values of the non-reciprocal parameter $\chi = \Lambda/\lambda$. Solid lines indicate the analytical prediction from Eq. \ref{['msd_split']}. Enhanced diffusivity is observed for $\Lambda/\lambda = -0.95$. Reference lines indicate the ballistic ($\propto t$) and diffusive ($\propto t^0$) regimes. (b) Effective diffusivity $\mathcal{D}_S^{\rm eff}$ of the tracer as a function of $\chi$. Giant diffusivity emerges near $\chi = -1$, with the solid line showing the theoretical form from Eq. \ref{['asymt_msd_tracer']}. Data points correspond to results from ABP simulations.